In the following, we consider the 2-to-1 mappings of the form (REF ) with \(s\)  being a Niho exponent. The 2-to-1 mappings of this type are connected to the polynomials
of the form \(h(x)=x^r\bar{h}(x^s)\)  over \( {\mathbb {F}}_{2^n}\) , where \(s \,|\, (2^n-1)\)  and \(\bar{h}[x] \in  {\mathbb {F}}_{2^n}[x]\) . It is known that the permutation property of the polynomials
of this form have been extensively studied in the literature. See, for example, [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}. To this end, we first give some preliminary lemmas.
